#8930461

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(35,41)(39,46)(42,44)(47,48);(2,4)(7,11)(8,12)(14,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(3,9)(5,13)(7,11)(8,12)(18,24)(19,25)(21,30)(28,31)(34,38)(35,41)(40,43)(42,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,31)(30,32)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20)(18,25)(19,24)(21,28)(22,23)(26,32)(27,29)(30,31)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837657);